Seismic analysis methods

Seismic waves propagate in both vertical and horizontal directions, and special attention should be paid to the horizontal vibrations because traditional calculation models do not consider this type of action in design. Vertical vibrations are less worrying, as they act in the direction of the structure's self-weight, which, in general, with design considering the proper calculation coefficients, sufficient safety is already provided.

According to Dantas (2013), the main parameters involved in seismic analyzes are the duration of the event and the predominant period, with fundamental importance in the non- linear analysis of structures, and the maximum values of the acceleration (PGA - Peak Ground Acceleration), velocity (PGV – Peak Ground Velocity) and displacement (PGD – Peak Ground Displacement), which depend on the characteristics of the fault distance, the nature of the soil formations traversed by the seismic waves and the local geological conditions. Regarding civil constructions, regulatory texts treat maximum acceleration as the most important parameter.

The response of structures to seismic action involves qualitative and quantitative parameters and can be evaluated through different approaches. Among the type of analysis, it

can be cited the linear elastic, nonlinear static, and nonlinear dynamic with time-integration, which may present modal characteristics and rely on the aid of numerical simulations.

It is necessary to determine the forces that represent the seismic action on the building and the consequences they produce before the effective design of the structural members. Most regulatory standards define seismic loading via the elastic acceleration response spectrum, Figure 63, equivalent to a 10% probability of being exceeded over a 50-year return period.

Figure 63: Response spectrum depending on the period.

(a) ABNT NBR 15421 (2006) (b) ASCE/SEI 7 (2016) Source: Adapted from ABNT NBR 15421 (2006) and ASCE/SEI 7 (2016).

Despite the development of advanced analysis tools, simplified methodologies are still preferred by engineers. In this context, most standards allow, for its better understanding, the method of equivalent lateral forces, in which total horizontal forces are applied at the base of the structure and on each floor for each of the main directions. These forces depend on the response modification factors, the system's total weight, spectral acceleration, and the structure's natural period.

In the method of equivalent horizontal forces, a reduced elastic response spectrum is considered by employing behavior coefficients, which allows for accounting for the inelastic capacity of the structure to dissipate energy through deformations and induced damage (Mohammadi and Naggar, 2004). The behavior coefficients vary depending on the building typology, and they are generally presented in the regulatory standards as the response modification, overstrength, and deflection amplification factors for each type of seismic force- resisting system.

Non-linear static analysis methods (pushover analysis) can be used to include inelastic parameters in safety assessments. These methods are based on controlling damage and deformation mechanisms for specific performance levels. Marques and Lourenço (2012)

clarifies that it is necessary to predict the capacity curve of buildings, which represents the relationship between the horizontal seismic force and the displacement of a significant control point of the structure. The curve is calculated by simulating an incremental static lateral load on the structure by assuming a uniform distribution of forces proportional to the inertia masses or modal distribution, in which seismic forces proportional to the inertia masses multiplied by the displacements of the first mode of vibration of the structure are used. The process is incremental and iterative, making its application difficult if computational resources are not used. It is indicated to use models with finite macro-elements, as illustrated in Figure 64, in which the damage progression on the panels can be observed, controlling the evolution of the capacity curve (Marques and Lourenço, 2012).

Figure 64: Structural masonry building pushover analysis.

Source: Adapted from Marques and Lourenço (2012).

In the dynamic analysis of structures, the equilibrium is governed by the Equation 49, considering the inertia forces dependent on the acceleration imposed (vector 𝑢̈) to the mass (matrix m) in each of the degrees of freedom, the forces on the elastic elements calculated by multiplying the stiffness (matrix k) by the displacements (vector 𝑢), and the viscous damping forces expressed as the product of the damping (matrix c) by the velocities (vector 𝑢̇). The balance is made by equating it to the product of the mass and the acceleration at the base of the building (vector 𝑠̈(𝑡)).

𝑚𝑢̈ + 𝑐𝑢̇ + 𝑘𝑢 = −𝑚𝑠̈(𝑡) Eq. 49 Depending on the desired precision level, the structure's dynamic analysis can be performed using different methods, varying the consideration of the inelastic behavior, the way of defining the seismic excitation, and the calculation procedure. In order to consider nonlinearity, it is necessary to modify the dynamic equilibrium equations, usually adopting a hysteretic rule simulating the cycles of loading, unloading, and reloading of the elements

Base shear

Displacement Uncracked Plastic by shear Shear failure Plastic by flexure Flexure failure

(Marques and Lourenço, 2012). According to Paulay and Priestly (1992), the inelastic time- integration method is considered one of the most sophisticated for predicting forces and displacements under seismic action, as it involves step-by-step resolution in the time domain of the equations of motion, considering multiple degrees of freedom to represent the response of a multi-story building. This analysis model is not usual because it involves complex concepts that designers find challenging to interpret and apply.